CAIE P3 (Pure Mathematics 3) 2015 June

1 Solve the equation \(\ln ( x + 4 ) = 2 \ln x + \ln 4\), giving your answer correct to 3 significant figures.

2 Solve the inequality \(| x - 2 | > 2 x - 3\).

3 Solve the equation \(\cot 2 x + \cot x = 3\) for \(0 ^ { \circ } < x < 180 ^ { \circ }\).

4 The curve with equation \(y = \frac { \mathrm { e } ^ { 2 x } } { 4 + \mathrm { e } ^ { 3 x } }\) has one stationary point. Find the exact values of the coordinates of this point.

5 The parametric equations of a curve are $$x = a \cos ^ { 4 } t , \quad y = a \sin ^ { 4 } t$$ where \(a\) is a positive constant.

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Show that the equation of the tangent to the curve at the point with parameter \(t\) is $$x \sin ^ { 2 } t + y \cos ^ { 2 } t = a \sin ^ { 2 } t \cos ^ { 2 } t$$
3. Hence show that if the tangent meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\), then $$O P + O Q = a$$ where \(O\) is the origin.

6 It is given that \(\int _ { 0 } ^ { a } x \cos x \mathrm {~d} x = 0.5\), where \(0 < a < \frac { 1 } { 2 } \pi\).

1. Show that \(a\) satisfies the equation \(\sin a = \frac { 1.5 - \cos a } { a }\).
2. Verify by calculation that \(a\) is greater than 1 .
3. Use the iterative formula $$a _ { n + 1 } = \sin ^ { - 1 } \left( \frac { 1.5 - \cos a _ { n } } { a _ { n } } \right)$$ to determine the value of \(a\) correct to 4 decimal places, giving the result of each iteration to 6 decimal places.

7 The number of micro-organisms in a population at time \(t\) is denoted by \(M\). At any time the variation in \(M\) is assumed to satisfy the differential equation $$\frac { \mathrm { d } M } { \mathrm {~d} t } = k ( \sqrt { } M ) \cos ( 0.02 t )$$ where \(k\) is a constant and \(M\) is taken to be a continuous variable. It is given that when \(t = 0 , M = 100\).

1. Solve the differential equation, obtaining a relation between \(M , k\) and \(t\).
2. Given also that \(M = 196\) when \(t = 50\), find the value of \(k\).
3. Obtain an expression for \(M\) in terms of \(t\) and find the least possible number of micro-organisms.

8 The complex number 1 - i is denoted by \(u\).

1. Showing your working and without using a calculator, express $$\frac { \mathrm { i } } { u }$$ in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
2. On an Argand diagram, sketch the loci representing complex numbers \(z\) satisfying the equations \(| z - u | = | z |\) and \(| z - \mathrm { i } | = 2\).
3. Find the argument of each of the complex numbers represented by the points of intersection of the two loci in part (ii).

9 Two planes have equations \(x + 3 y - 2 z = 4\) and \(2 x + y + 3 z = 5\). The planes intersect in the straight line \(l\).

1. Calculate the acute angle between the two planes.
2. Find a vector equation for the line \(l\).

10 Let \(\mathrm { f } ( x ) = \frac { 11 x + 7 } { ( 2 x - 1 ) ( x + 2 ) ^ { 2 } }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Show that \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 4 } + \ln \left( \frac { 9 } { 4 } \right)\).